2006 fall math 100 lecture 141 math 100 class 20 line integral
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2006 Fall MATH 100 Lecture 14 1
MATH 100 Class 20 Line Integral
.or respect to with along or of integral line
)('))(),((),(
)('))(),((),(
then0, containingregion open some
on continuous be y)g(x,&y)f(x,let and
b) ~ t ~ (a y(t) y x(t), x
space-2in curvesmooth a be Clet :Definition
yxCgf
dttytytxgdyyxg
dttxtytxfdxyxf
b
aC
b
aC
2006 Fall MATH 100 Lecture 14 2
MATH 100 Class 20 Line Integral
form. aldifferenti-
),(),(),(),(
integral line combined the
with deal toneed wen,applicatioIn
gdyfdx
dyyxgdxyxfdyyxgdxyxfC CC
2006 Fall MATH 100 Lecture 14 3
MATH 100 Class 20 Line Integral
)2t(0sin t Y t,cos xcircular over
)(2 :ExC
22
dyyxxydx
2006 Fall MATH 100 Lecture 14 4
MATH 100 Class 20 Line Integral
1sincos1)(
3
22
)(sinsin2)sin(sincos2
2
22
22
00
22
1
0
2
0
2
0
ttdtdyyx
udu
ttddtttt
xydx
C
C
:Sol
2006 Fall MATH 100 Lecture 14 5
MATH 100 Class 20 Line Integral
3
11
3
2)(2
C
22 dyyxxydx
Remark:
1. Independence of parameterization
20,sincos
40,2sin2cos
20,sincos
22
ttytx
ttytx
ttytx
all produce 1/3
2006 Fall MATH 100 Lecture 14 6
MATH 100 Class 20 Line Integral
2. Reversal of orientationIf we reverse the orientation of the line integral, the line integral is the negation of the original result.
10,1
20,2
sin,2
cos
223),sin(),cos(
2
ttytx
ttytx
ttytx
all produce -1/3
2006 Fall MATH 100 Lecture 14 7
MATH 100 Class 20 Line Integral
3. let -C denote C with reverse orientation when
gdyfdxgdyfdxCC
line integral over piecewise smooth curve Figure 18.1.2
kCCCC
...21
2006 Fall MATH 100 Lecture 14 8
MATH 100 Class 20 Line Integral
C
xdyydxx2 :Ex
(0,0)(1,2)(1,0)(0,0) gleover trian
2006 Fall MATH 100 Lecture 14 9
MATH 100 Class 20 Line Integral
21)1(1
20,,1:
000
10,0,:
22
11
22
2
22
1
CC
CC
dttdxdyydxx
ttyxC
tddttxdyydxx
tytxC
:Sol
2006 Fall MATH 100 Lecture 14 10
2
1
2
320
2
3
2
1
4
12)(2
22
01:,2,:
2
1
0
3
0
1
20
1
22
3
3
C
C
xdyydxx
dttt
tdttdttxdyydxxxdyydxx
ttytxC
MATH 100 Class 20 Line Integral
2006 Fall MATH 100 Lecture 14 11
Vector notation: let jyixr
jyxgiyxfyxF
),(),(),( and
jdyidxrd
dyyxgdxyxfrdyxF ),(),(),(
MATH 100 Class 20 Line Integral
2006 Fall MATH 100 Lecture 14 12
For parametric expression of curve
dtdt
rddtj
dt
dyi
dt
dxjdt
dt
dyidt
dt
dxtrd
jtyitxr
)(
)()(
dtdt
dytytxg
dt
dxtytxfdt
dt
rdyxF
rdyxF
))(),(())(),((),(
),(
MATH 100 Class 20 Line Integral
2006 Fall MATH 100 Lecture 14 13
and
b
a
b
a
C
dtdt
dytytxg
dt
dxtytxf
dtdt
rdtytxF
rdyxF
))(),(())(),((
))(),((
),(
MATH 100 Class 20 Line Integral
2006 Fall MATH 100 Lecture 14 14
motion ofdirection in the Force ofcomponent cos
avelleddistant tr-
cos
is work then the, to from linestraight a
alongmay partial aon nets force work ofn calculatio
:meaning Physical
F
PQ
PQFPQFW
QP
F
MATH 100 Class 20 Line Integral
2006 Fall MATH 100 Lecture 14 15
What about varying force along a smooth curve? Figure 18.1.4
Let bxajtyitxtr
)()()(
MATH 100 Class 20 Line Integral
jyxgiyxfyxF
),(),(),(
denote the curve and
denote the force
2006 Fall MATH 100 Lecture 14 16
MATH 100 Class 20 Line Integral
)()(
:)( to)( from moves
on donework
Then
)(:)( to)( from moves on donework
)(:)( to)( from moves on donework
denote also
twttw
trttr
PQ
ttwttrarPQ
twtrarPQ
thusconstant, a toclose &
line,straight a toclose is
)( to)( then ,small is If
F
trttrt
2006 Fall MATH 100 Lecture 14 17
MATH 100 Class 20 Line Integral
rdyxFdttrtytxF
dttwawbwba
aw
trtytxFtw
t
trttrtytxF
t
twttw
trttrtytxFtwttw
C
),( )('))(),((
)( )()( to from doneWork
0 )( that Note
)('))(),(()('
)()())(),((lim
)()(lim and
)]()())[(),(( )()(
b
a
b
a
0tt
2006 Fall MATH 100 Lecture 14 18
MATH 100 Class 20 Line Integral
34
2
3
2
6
1
)](2[]21[)(
))(),((),()(
)()())(),((
12)(
12,
)(),(
along )1,1( to)4,2( from:
1
2
436
1
2
251
2
25
25223
2
2
3
2
ttt
dtttttdtjtijttit
dtdt
rdtytxFrdyxFbw
jttitjttitttytxF
tjtittr
ttytx
jyxiyxyxF
xyC
b
aC
2006 Fall MATH 100 Lecture 14 19
MATH 100 Class 20 Line Integral
b
aC
b
aC
b
aC
CCC
C
dttztztytxhdxzyxh
dttytztytxgdxzyxg
dttxtztytxfdxzyxf
dzzyxhdyzyxgdxzyxf
dzzyxhdyzyxgdxzyxf
C
Czyxhzyxgzyxf
btatzztyytxx
)('))(),(),((),,(
)('))(),(),((),,(
)('))(),(),((),,( where
),,(),,(),,(
),,(),,(),,(
as defined is along integral linethen
,contain region on continuous are ),,(),,,(),,,(
)(),(),(
space-3in integral Line
2006 Fall MATH 100 Lecture 14 20
MATH 100 Class 20 Line Integral
C
CC
),,( C alongwork
))(),(),((),,(then
))()()(
(
)()()()(
),,(),,(),,(),,(Let
rdzyxFw
tddt
rdtztytxFrdzyxFhdzgdyfdx
dtkdt
tdzj
dt
tdyi
dt
tdxdt
dt
rdrd
ktzjtyitxtr
kzyxhjzyxgizyxfzyxF
b
a
C
),,( rdzyxF
2006 Fall MATH 100 Lecture 14 21
MATH 100 Class 20 Line Integral
66)32(
)32()(
))(),(),((
:Sol
),,(
1t0 ,)( :
:Ex
1
0
51
0
555
1
0
2345
C
1
0
32
tdttdttt
tdktjtiktjtit
tddt
rdtztytxFrdF
kxyjxziyzzyxF
ktjtittrC